Does time have a minimum 'speed'? Sorry if this is an ignorant question, but I've been having some trouble grasping some concepts related to time dilation. So far, my understanding of the concept says that if I am in a certain frame of reference, and another frame of reference is moving at let's say 0.99c, there is a very evident effect of time dilation where time passes much slower in frame 2 compared to frame 1. And we have observed relativistic effects in satellites etc. because of their higher speed of motion with respect to Earth.
If I move away from the gravitational effects of Earth, and am able to observe Earth spin away from me, the same theory would hold for a clock sitting on Earth versus the clock in my hand. But if I move away from every possible cause of motion (galaxies moving from expansion etc.), and am absolutely still in space: what would happen to the clock in my hand? The whole thought behind relativity is the idea of nothing being absolute, so that being the case, to what extent can I 'decelerate' time in my frame?
 A: It does not matter whether you are moving relative to anything else that you can observe or just in a fixed reference frame in a completely empty space. Special relativity treats both scenarios as the same because you experience no acceleration/gravity. 
No matter what state of motion you are in your time will pass with the same speed as always so in a sense the speed of time never changes for one observer. It is only that relative to each other the clocks dilate. 
A good analogy for this is bond enthalpy. It cannot be an absolute quantity. One can only determine the change in bond enthalpy during a reaction in chemistry.
Similarly time is not absolute but to be viewed in respect to another time frame. 
A: We do not have an absolute rate for time flow. So, slow or fast depend in what reference frame you are fixing to define what is fast and what is slow.
What make sense is asking if we can make the rate of time flow slowly as possible when compared with the proper time for example. And the answer to this question is yes. Any referential frame with constant acceleration will see an clock in inertia freezing at some instant of time. See the Rindler observer.
